Question: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 8x - 64$, and $ m \angle ABC = 9x - 101$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {9x - 101} + {8x - 64} = {90}$ Combine like terms: $ 17x - 165 = 90$ Add $165$ to both sides: $ 17x = 255$ Divide both sides by $17$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 9({15}) - 101$ Simplify: $ {m\angle ABC = 135 - 101}$ So ${m\angle ABC = 34}$.